166944.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

10 declarations, 1 contain sorry, 0 declared as axioms

Rowland-style prime-generating recurrence

The sequence is defined by $a(1) = 2$ and for $n \ge 2$: $a(n) = a(n-1) + \gcd(n, a(n-1))$ if $n$ is even, and $a(n) = a(n-1) + \gcd(n-2, a(n-1))$ if $n$ is odd.

References:

A166944
E. S. Rowland, "A natural prime-generating recurrence", J. Integer Sequences 11 (2008), Article 08.2.8.
V. Shevelev, "An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes", arXiv preprint arXiv:0910.4676 [math.NT], 2009.

Imports (1)

FormalConjecturesUtil

def OeisA166944.a Defining recurrence for $a(n)$ line 40 source
Source documentation
Defining recurrence for $a(n)$.

no axioms what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℕ
  | 0 => 0
  | 1 => 2
  | n + 2 =>
    let prev := a (n + 1)
…
```
theorem OeisA166944.a_1 Value of the sequence a at 1 line 51 source
Source documentation
Value of the sequence a at 1.

no axioms what this means
Statement uses: Eq, Nat, OeisA166944.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 2 := by decide
…
```
Mentions: OeisA166944.a
theorem OeisA166944.a_2 Value of the sequence a at 2 line 55 source
Source documentation
Value of the sequence a at 2.

no axioms what this means
Statement uses: Eq, Nat, OeisA166944.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 4 := by decide
…
```
Mentions: OeisA166944.a
theorem OeisA166944.a_3 Value of the sequence a at 3 line 59 source
Source documentation
Value of the sequence a at 3.

no axioms what this means
Statement uses: Eq, Nat, OeisA166944.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 5 := by decide
…
```
Mentions: OeisA166944.a
theorem OeisA166944.a_4 Value of the sequence a at 4 line 63 source
Source documentation
Value of the sequence a at 4.

no axioms what this means
Statement uses: Eq, Nat, OeisA166944.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 6 := by decide
…
```
Mentions: OeisA166944.a
theorem OeisA166944.a_5 Value of the sequence a at 5 line 67 source
Source documentation
Value of the sequence a at 5.

no axioms what this means
Statement uses: Eq, Nat, OeisA166944.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 9 := by decide
```
Mentions: OeisA166944.a
def OeisA166944.d The difference sequence $d(n) = a(n) - a(n-1)$ for $n \ge 2$ line 70 source
Source documentation
The difference sequence $d(n) = a(n) - a(n-1)$ for $n \ge 2$.

no axioms what this means
Statement uses: Nat
Proof uses: HSub.hSub, OeisA166944.a, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
def d (n : ℕ) : ℕ := a n - a (n - 1)
```
Mentions: OeisA166944.a
def OeisA166944.IsGreaterTwinPrime A prime $p$ is the greater of a twin prime pair if $p$ and $p - 2$ are both prime line 73 source
Source documentation
A prime $p$ is the greater of a twin prime pair if $p$ and $p - 2$ are both prime.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: And, Nat.Prime, HSub.hSub, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
def IsGreaterTwinPrime (p : ℕ) : Prop := p.Prime ∧ (p - 2).Prime
```
def OeisA166944.IsDifferenceRecord A value $R$ is a difference record if there exists $n \ge 2$ such that $d(n) = R$ and $R$ is strictly larger than all previous differences $d(k)$ for $2 \le k < n$ line 78 source
Source documentation
A value $R$ is a difference record if there exists $n \ge 2$ such that $d(n) = R$ and $R$ is strictly larger than all previous differences $d(k)$ for $2 \le k < n$.

no axioms what this means
Statement uses: Nat
Proof uses: Exists, And, LE.le, OfNat.ofNat, Eq, OeisA166944.d, LT.lt
3 instances, coercions and auxiliary constants left out.
```
def IsDifferenceRecord (R : ℕ) : Prop :=
…
```
theorem OeisA166944.conjecture Conjecture: Every record of differences $a(n)-a(n-1)$ more than 5 is the greater of twin primes (A006512) sorry line 85 source
Source documentation
Conjecture: Every record of differences $a(n)-a(n-1)$ more than 5 is the greater of twin primes (A006512).

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, OeisA166944.IsDifferenceRecord, OeisA166944.IsGreaterTwinPrime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (R : ℕ) (hR : 5 < R) (hrec : IsDifferenceRecord R) :
    IsGreaterTwinPrime R := by
…
```
Mentions: OeisA166944.IsDifferenceRecord, OeisA166944.IsGreaterTwinPrime